Maximum weighted induced forests and trees: New formulations and a computational comparative review

02/18/2021 ∙ by Rafael A. Melo, et al. ∙ Universidade Federal Fluminense UFBA em Pauta 0

Given a graph G=(V,E) with a weight w_v associated with each vertex v∈ V, the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset V'⊆ V of the vertices such that V' induces a forest. This NP-hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem (MWIT), on the other hand, requires that the subset V'⊆ V induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch-and-cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial mixed integer programming solver. More specifically, five formulations are compared, two compact (i.e., with a polynomial number of variables and constraints) ones and the three others with an exponential number of constraints. The experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the small instances, especially the more challenging ones. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to compare the optimal solution values of the two problems.

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1 Introduction

Graph theory problems related to encountering induced subgraphs with certain properties have been extensively studied in the literature due to their theoretical interest and various practical applications. The maximum weighted induced forest problem (MWIF) belongs to this category of problems. Let us consider a simple and undirected graph with a set of vertices and a set of edges, and a nonnegative weight associated with each vertex . For any subset of the vertices, we denote by the graph induced in by , whose edge set is formed by all edges in with both extremities in , i.e., . The problem thus consists of finding a maximum weighted subset inducing a forest , i.e., an acyclic induced subgraph. The maximum weighted induced tree problem (MWIT) consists of obtaining a maximum weighted subset inducing a tree , i.e., an acyclic and connected induced subgraph. Maximum weighted induced trees and forests are illustrated in Figure 1.

Figure 1: Illustration of maximum weighted induced trees and forests. Subfigure (a) exemplifies an input graph with seven nodes and . Node weights are represented outside nodes. Subfigure (b) pictures a maximum weighted induced tree , with and weight 30. Subfigure (c) exemplifies a maximum weighted induced forest , with and weight 33.

Given a graph , a feedback vertex set is a subset of its vertices whose removal results in an acycic graph. Note, thus, that induces a forest. In this regard, induced forest problems have appeared in several applications in the literature mainly via feedback vertex set problems, which are their equivalent complementary counterparts. These include preventing and removing deadlocks in operating systems Wang et al. (1985); Carneiro et al. (2019), program verification Shamir (1979), constraint satisfaction Dechter and Pearl (1988), study of monopolies in distributed systems Peleg (1998)

, bayesian inference 

Bar-Yehuda et al. (1998), combinatorial circuit design Bafna et al. (1999), optical networks Kleinberg and Kumar (2001), parallel systems and distributed computing Bossard (2014), and automated storage/retrieval systems Gharehgozli et al. (2021).

Several authors have considered theoretical studies and approaches for induced forests and feedback vertex sets. Bei97 considered the problem of decycling a graph and introduced the concept of the decycling number of a graph, given by the minimum number of vertices to be removed so that the resulting graph is acyclic (i.e., the size of a minimum cardinality feedback vertex set). BruMafTru00 conducted a polyhedral study and proposed a tabu search metaheuristic for the minimum weighted feedback vertex set problem (MWFVS). CarCerGenPar05 proposed a linear time algorithm for the MWFVS on a special class of “diamond” graphs, as called by the authors. CarCerGen11 implemented a tabu search metaheuristic for the MWFVS which makes use of the algorithms described in CarCerGenPar05 in the local search procedure. CarCerCer14 proposed a memetic algorithm for the MWFVS. ShiXu17 demonstrated graph theoretical results regarding the existence of induced forests with a certain number of vertices. Very recently, MelQueRib21 tackled the MWFVS via the MWIF and proposed compact integer programming formulations and an interated local search (ILS) based matheuristic for the problem. A review of the literature on feedback vertex set problems has previously appeared in FesParRes99.

To the best of our knowledge, the majority of the works related to encountering induced trees in graphs have concentrated on graph theoretical and computational complexity aspects. ErdSakSos86 considered the maximum cardinality induced tree problem and provided bounds for its size which are related to other parameters of the graph. Sco97 analyzed the existence of induced trees in graphs with large chromatic number. Rau07 studied dominating sets inducing trees and provided bounds on the maximum number of vertices of induced trees for certain classes of graphs. DerPic09 considered complexity results for problems related to encountering induced trees covering a prescribed set of vertices. ChuSey10 presented a polynomial time algorithm to decide whether there is an induced tree containing three given vertices of a graph.

Other problems of obtaining connected or induced subgraphs, which are somehow related to the problems considered in our work, were also studied in the recent literature. LjuWeiPfeKlaMutFis06, CosCorLap09, and SieAhmNem20 proposed integer programming approaches for variants of the Steiner tree problem. The maximum weight connected subgraph problem has been studied by several authors Álvarez-Miranda et al. (2013); Rehfeldt and Koch (2019); Liu et al. (2020). MarPar16 considered the problem of encountering a maximum balanced induced subgraph. AgrDahHauPin17 tackled the problem of finding maximum -regular induced subgraphs of a graph. MatVerProPas19, BokChiWagWie20, and MarRib21 considered the problem of encountering the longest induced path. Besides, integer programming approaches have been successfully applied to several optimization problems related to encountering trees and forests with certain properties Melo et al. (2016); Carrabs et al. (2018, 2021).

The main contributions of this work can be summarized as follows. Firstly, we propose two new formulations for the maximum weighted induced forest problem together with branch-and-cut procedures. Secondly, we perform extensive computational experiments comparing the proposed formulations with others available in the literature. Thirdly, we show that the proposed formulations can be easily adapted to solve the maximum weighted induced tree problem, which, to the best of our knowledge, stand as the first optimization approaches for the problem.

The remainder of this paper is organized as follows. Section 2 describes the integer programming formulations for the maximum weighted induced forest problem. Section 3 details the proposed branch-and-cut approaches. Section 4 shows a simple way to adapt the proposed formulations to deal with the maximum weighted induced tree problem. Section 5 reports the computational experiments. Concluding remarks are discussed in Section  6.

2 Formulations

In this section, we propose two new formulations for the maximum weighted induced forest problem (MWIF), and describe three previously existing formulations Brunetta et al. (2000); Melo et al. (2021). Similarly to MelQueRib21, the input graph for MWIF is slightly transformed such that the problem consists in encountering a tree in this modified graph. Given the input graph , an alternative transformed graph is obtained with vertex set and edge set . The weights of all vertices in remain the same in , while the weight of the new vertex is . The dummy vertex is used to join the connected components of the induced forest .

2.1 Cycle elimination formulation

The first described formulation does not have variables corresponding to the edges and is the only one described in this work which does not use the transformed graph. It is equivalent to the formulation proposed in BruMafTru00 for the minimum weighted feedback vertex set problem, with the only difference being that the variables represent the vertices in the induced forest rather than in the feedback vertex set. Let be the family of all subsets such that induces a cycle. Define the variable to be equal to one if vertex is in the induced forest , otherwise. A cycle elimination-based formulation can be obtained as

(1)
(2)
(3)

The objective function (1) maximizes the sum of the weights of the vertices in the forest, i.e., those such that . Constraints (2) ensure the elimination of cycles. Constraints (3) guarantee the integrality of the variables.

2.2 Tree with cycle elimination formulation

This new undirected formulation considers the modified graph described at the beginning of this section. It guarantees the solution to be acyclic similarly to the cycle elimination formulation. Define the variable to be equal to one if vertex is in the induced forest, otherwise. Additionally, consider variable to be equal to one if edge is in the induced forest, otherwise. The tree with cycle elimination formulation can be obtained as

(1 revisited)
(4)
(2 revisited)
(5)
(6)
(7)
(8)
(9)

Constraint (4) ensures the number of edges in the solution equals the number of selected vertices. Constraint (5) forces the dummy node to be in the solution. Constraints (6) and (7) force the resulting graph to be induced. Note that constraints (6) guarantee that an edge can only be in the solution if both its endpoints are in the solution, while constraints (7) ensure that if both endpoints of an edge are in the solution, then the edge must also be in the solution. Constraints (8) and (9) determine the integrality of the and variables, correspondingly.

2.3 Flow-based formulation

The flow-based formulation Melo et al. (2021), as well as the two formulations described next in Sections 2.4 and 2.5, considers a directed version of graph , which we denote . The set is obtained by creating arcs and for each edge . Denote by and , respectively, the sets of arcs outgoing from and incoming into vertex . A flow-based formulation modeling a tree can be obtained by sending one unit of flow from the dummy vertex to each vertex included in the solution inducing a forest. Define variable to represent the flow going from vertex to vertex , for every arc . Let variable be defined as before for every , and consider variable to be equal to one if arc is in the solution, otherwise. Thus, a flow-based formulation Melo et al. (2021) can be defined as

(1 revisited)
(10)
(11)
(12)
(13)
(5 revisited)
(14)
(15)
(16)
(17)
(18)

Constraints (10) ensure there is exactly one arc incoming at each selected vertex . Balance constraints (11) and (12) are flow conservation constraints for each vertex. Constraints (13) link the flow variables with the variables. Similarly to constraints (6) and (7), constraints (14) and (15) establish that the solution is an induced subgraph . Constraints (16), (17), and (18) ensure the integrality and nonnegativity of the corresponding variables.

As mentioned in MelQueRib21, one could also derive a multicommodity flow based formulation Magnanti and Wolsey (1995) for MWIF, but such type of formulation is usually not viable in practice for large problem instances as even solving its linear relaxation can be challenging Costa et al. (2009); Carrabs et al. (2013).

2.4 Miller-Tucker-Zemlin-based formulation

The Miller-Tucker-Zemlin-based (MTZ-based) formulation for MWIF Melo et al. (2021) also considers the directed graph . It adapts the formulation of CosCorLap09 which applies the MTZ constraints Miller et al. (1960); Desrochers and Laporte (1991) to solve a variant of the Steiner tree problem. Define variables and as previously. Additionally, let be a potential variable associated with each vertex . An MTZ-based formulation Melo et al. (2021) can thus be cast as

(1 revisited)
(10 revisited)
(19)
(20)
(21)
(5 revisited)
(14 revisited)
(15 revisited)
(16 revisited)
(17 revisited)
(22)

Constraints (19) guarantee that if an arc is in the solution, then vertex has a potential value larger than that of , given by . Constraints (20) determine that if a vertex is in the solution, then its potential is neither zero nor greater than . Constraint (21) sets the potential of the dummy vertex to zero. Constraints (22) ensure the nonnegativity of the potential variables.

2.5 Directed cutset formulation

This new directed cutset formulation also considers the directed graph and guarantees the elimination of cycles using connectivity constraints. A feasible solution is an arborescence rooted at the dummy vertex and, for any partition with and for any vertex , it ensures that if is in the solution, at least one arc in the cut must be in the solution. Such approach was already applied in the context of Steiner trees Ljubić et al. (2006); Costa et al. (2009). Considering variables and as defined in Section 2.3, a directed cutset formulation can be described as

(1 revisited)
(10 revisited)
(23)
(5 revisited)
(14 revisited)
(15 revisited)
(16 revisited)
(3 revisited)

Constraints (23) ensure that whenever a vertex is in the solution, then there is an arc crossing the cut , with and .

3 Branch-and-cut approaches

This section describes the clique inequalities, which are used to strengthen the formulations, and details the separation procedures for the constraints which are exponential in number.

3.1 Clique inequalities

The clique inequalities were proposed for the minimum weighted feedback vertex set problem in BruMafTru00, and can be defined considering the variables described in our work as follows. Consider to be any clique, i.e., a complete induced subgraph, with . The inequalities

(24)

are valid for the maximum weighted induced forest problem. In fact, more than two vertices of a clique in a solution would induce a cycle.

3.2 Separation of cycle constraints

The separation of cycle constraints (3) takes as input a separation graph induced by the vertices corresponding to the nonzero variables in the solution, i.e., . The separation for integer solutions is performed using a modified depth-first search (DFS) algorithm Cormen et al. (2009) in

in which, for every back edge traversed in the search, the corresponding cycle is stored. After the DFS is finished, all the cycles encountered during its execution are provided to the solver. The separation for fractional solutions is performed heuristically using a DFS algorithm in

that considers the vertices in nonincreasing order of the corresponding values. For every back edge traversed during the search, the encountered cycle is checked for violation of (3) and in an affirmative case, it is stored. After the DFS is finished, all the cycles encountered during the search are given to the solver.

3.3 Separation of connectivity constraints

The separation of connectivity constraints (23) also takes as input a separation graph induced by the vertices corresponding to the nonzero variables. The separation for integer solutions is performed by a breadth-first search (BFS) algorithm Cormen et al. (2009) in , which initiates at the dummy vertex and composes the part of partition with the vertices in the same connected component of . Next, for each vertex in an inequality is generated and stored. All the encountered violated inequalities are provided to the solver at the end of the procedure. Separation for fractional solutions is performed exactly using maximum flows (minimum cuts) according to the approach described in MagWol95. Basically, a new directed graph is built from the solution , where the capacities of the arcs are given by the values of . A maximum flow (minimum cut) problem is solved from to each . In case a constraint (23) is found to be violated, it is stored. All the encountered violated inequalities are provided to the solver at the end of the procedure.

3.4 Separation of clique inequalities

The separation of clique inequalities (24) is performed heuristically, and also takes as input a separation graph induced by the vertices corresponding to the nonzero variables. The heuristic orders the vertices in in nonincreasing order of the corresponding values and in case of ties, in nonincreasing order of degree in . The heuristic, then, greedly chooses a vertex which is adjacent to all other vertices already chosen. This procedure is repeated until all vertices are in a maximal clique in . Whenever a violated clique is encountered, an attempt to lift it is performed. This is achieved by turning it into a maximal clique in the original graph , whenever possible, via the insertion of vertices which were not in the separation graph. This step uses a similar greedy idea to the one used to iteratively build the clique in the separation graph. The difference lies in the fact that it only considers the degree of the vertices as a greedy criterium.

4 Adapting the formulations to solve the maximum weighted induced tree problem

We devote this section to a simple, but relevant observation. Namely, that the tree with cycle elimination, flow-based, MTZ-based and directed cutset formulations can be directly adapted to tackle the maximum weighted induced tree problem.

This can be achieved with the addition of a single constraint. More specifically, the undirected tree with cycle elimination formulation can ensure that the solution is an induced tree by adding the constraint

(25)

while the directed flow-based, MTZ-based and cutset formulations can guarantee that the solution is an induced tree by adding

(26)

We note, however, that these adaptations will not be used in the computational experiments.

5 Computational experiments

In this section, we summarize the computational experiments performed to assess the effectiveness of the formulations described in Section 2. All computational experiments were carried out on a machine running under Ubuntu GNU/Linux, with an Intel(R) Core(TM) i7-4770 CPU @ 3.40GHz processor and 16Gb of RAM. The algorithms were coded in Julia v1.4.2, using JuMP v0.18.6. The formulations were solved using Gurobi 9.0.2.

5.1 Instances

The experiments were performed using the benchmark instances proposed in CarCerGen11 for the maximum weighted feedback vertex set problem, where more details can be encountered. The instances are available in instances. They consist of random, grid, toroidal, and hypercube graphs, and are classified as small, with

, or large, with

. The node weights are uniformly distributed in one of the following intervals:

, , or . Similar instances, but with different weights, are organized into instance groups containing five instances each. Each instance group is identified as , where gives the class of the graph: square grid (), non-square grid (), hypercube (), toroidal (), or random (). For random graphs and denote, respectively, the numbers of vertices and edges; for grid and toroidal graphs and denote, correspondingly, the number of lines and columns of the grid; for hypercube graphs identifies the corresponding -hypercube graph; the other parameters indicate the lower () and upper () bounds on the weights. We remark that the results reported throughout this section represent average values over the five instances belonging to the same instance group.

5.2 Tested approaches and settings

The following approaches were considered in our experiments:

  • cycle elimination formulation (CYC) strengthened with clique inequalities;

  • tree with cycle elimination formulation (TCYC) strengthened with clique inequalities;

  • compact flow-based formulation (FLOW);

  • compact MTZ-based formulation (MTZ); and

  • directed cutset formulation (DCUT) strengthened with clique inequalities.

The MIP solver was executed using the standard configurations, except the relative optimality tolerance gap which was set to . A time limit of 3600 seconds was imposed for each execution of the MIP solver. All the separation procedures were implemented as callbacks in the MIP solver. The separations for fractional solutions were only executed at the root node. Each formulation was tested (a) as a standalone approach and (b) receiving as input a good quality warm start (i.e., a feasible solution) generated by the matheuristic MILS- described in Section 3.6 of MelQueRib21.

We remark that BruMafTru00 also showed that certain subset inequalities are valid for the feedback vertex set problem. However, the authors observed in the computational experiments that these inequalities slowed down the solution process and, for this reason, we did not use such inequalities in our experiments.

5.3 Small instances

Tables 1-10 summarize the results for the small instances, with Tables 1-5 corresponding to the formulations as standalone approaches and Tables 6-10 to the formulations when offered warm starts. In each of these tables, the first column identifies the instance group. Next, for each of the approaches (CYC, TCYC, FLOW, MTZ, and DCUT), the tables show the average best value (avg), the number of instances solved to optimality (opt) followed by the average time to solve them (time), and the average relative open gap (gap) in percent considering the instances which were not solved to optimality, given by , where gives the best bound and the best solution value. Whenever the majority (i.e., at least three) of the formulations are able to solve the same amount of instances, the best average time is shown in bold and blue. Otherwise, the maximum number of instances solved to optimality appears in bold and blue and, in case no instances were solved to optimality by any of the formulations, the smallest gap is represented in bold and blue.

Tables 1 and 2 show that all small grid instances could be solved to optimality using all the five formulations, most of them in low computational times. It can be noted that FLOW achieved the lowest average times for most of the instance groups, while DCUT needed more time on average to solve these instances. Table 3 shows that, for the small hypercube instances, all formulations but CYC were able to solve all instances to optimality. TCYC achieved the lower average times. It can also be noted that the average times for FLOW and MTZ are very similar for these instances. Table 4 shows that, for the small toroidal instances, all formulations but CYC solved all instances to optimality, with FLOW achieving lower average times. Table 5 shows that for the small random instances, again, all formulations but CYC were able to solve all of them to optimality, with TCYC achieving lower average times, followed by DCUT. It can be seen that, for these instances, TCYC and DCUT outperformed by far MTZ and FLOW, especially for the larger instances in the subset, indicating the benefits of a branch-and-cut approach using the formulations with clique inequalities.

CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
G_5_5_10_25 352.4 5 1.4 352.4 5 1.1 352.4 5 0.1 352.4 5 0.1 352.4 5 1.7
G_5_5_10_50 636.8 5 1.4 636.8 5 1.1 636.8 5 0.1 636.8 5 0.1 636.8 5 1.8
G_5_5_10_75 934.2 5 1.4 934.2 5 1.1 934.2 5 0.1 934.2 5 0.1 934.2 5 1.8
G_7_7_10_25 712.4 5 1.5 712.4 5 1.2 712.4 5 0.3 712.4 5 0.7 712.4 5 2.4
G_7_7_10_50 1324.2 5 1.5 1324.2 5 1.1 1324.2 5 0.3 1324.2 5 0.3 1324.2 5 2.1
G_7_7_10_75 1998.6 5 1.5 1998.6 5 1.2 1998.6 5 0.3 1998.6 5 0.4 1998.6 5 2.1
G_9_9_10_25 1180.8 5 6.3 1180.8 5 1.8 1180.8 5 2.3 1180.8 5 10.1 1180.8 5 177.3
G_9_9_10_50 2192.2 5 2.5 2192.2 5 1.3 2192.2 5 1.4 2192.2 5 2.8 2192.2 5 249.5
G_9_9_10_75 3452.2 5 2.4 3452.2 5 1.3 3452.2 5 1.1 3452.2 5 1.4 3452.2 5 86.7
Average 1420.4 2.2 1420.4 1.2 1420.4 0.7 1420.4 1.8 1420.4 58.4
Total 45 45 45 45 45
Table 1: Results comparing the formulations for the small square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
GNQ_8_3_10_25 319.2 5 1.4 319.2 5 1.1 319.2 5 0.1 319.2 5 0.1 319.2 5 1.8
GNQ_8_3_10_50 579.2 5 1.4 579.2 5 1.1 579.2 5 0.0 579.2 5 0.1 579.2 5 1.8
GNQ_8_3_10_75 838.0 5 1.3 838.0 5 1.1 838.0 5 0.0 838.0 5 0.1 838.0 5 1.8
GNQ_9_6_10_25 786.8 5 1.7 786.8 5 1.2 786.8 5 0.4 786.8 5 2.3 786.8 5 2.3
GNQ_9_6_10_50 1438.6 5 1.5 1438.6 5 1.2 1438.6 5 0.4 1438.6 5 0.7 1438.6 5 16.2
GNQ_9_6_10_75 2195.6 5 1.5 2195.6 5 1.2 2195.6 5 0.4 2195.6 5 0.6 2195.6 5 4.5
GNQ_12_6_10_25 1071.2 5 2.8 1071.2 5 1.3 1071.2 5 1.2 1071.2 5 5.0 1071.2 5 38.0
GNQ_12_6_10_50 1944.2 5 1.9 1944.2 5 1.3 1944.2 5 1.1 1944.2 5 2.8 1944.2 5 250.9
GNQ_12_6_10_75 3022.4 5 1.7 3022.4 5 1.2 3022.4 5 0.8 3022.4 5 1.8 3022.4 5 32.8
Average 1355.0 1.7 1355.0 1.2 1355.0 0.5 1355.0 1.5 1355.0 38.9
Total 45 45 45 45 45
Table 2: Results comparing the formulations for the small non-square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
I_4_10_25 147.0 5 1.4 147.0 5 1.1 147.0 5 0.0 147.0 5 0.1 147.0 5 1.8
I_4_10_50 289.4 5 1.4 289.4 5 1.1 289.4 5 0.0 289.4 5 0.0 289.4 5 1.8
I_4_10_75 344.0 5 1.4 344.0 5 1.1 344.0 5 0.0 344.0 5 0.0 344.0 5 1.7
I_5_10_25 278.4 5 2.3 278.4 5 1.3 278.4 5 0.8 278.4 5 0.7 278.4 5 2.8
I_5_10_50 545.0 5 1.5 545.0 5 1.2 545.0 5 0.2 545.0 5 0.2 545.0 5 1.9
I_5_10_75 787.4 5 1.5 787.4 5 1.1 787.4 5 0.1 787.4 5 0.1 787.4 5 1.9
I_6_10_25 527.2 3 2871.4 3.1 527.2 5 23.9 527.2 5 66.3 527.2 5 67.1 527.2 5 1767.4
I_6_10_50 975.6 5 22.5 975.6 5 2.8 975.6 5 5.2 975.6 5 4.2 975.6 5 152.0
I_6_10_75 1250.8 5 7.1 1250.8 5 1.4 1250.8 5 1.0 1250.8 5 0.7 1250.8 5 3.2
Average 571.6 323.4 3.1 571.6 3.9 571.6 8.2 571.6 8.1 571.6 215.0
Total 43 45 45 45 45
Table 3: Results comparing the formulations for the small hypercube instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
T_5_5_10_25 232.8 5 1.5 232.8 5 1.1 232.8 5 0.1 232.8 5 0.1 232.8 5 1.8
T_5_5_10_50 357.2 5 1.4 357.2 5 1.2 357.2 5 0.1 357.2 5 0.1 357.2 5 1.8
T_5_5_10_75 448.6 5 1.4 448.6 5 1.1 448.6 5 0.1 448.6 5 0.1 448.6 5 1.8
T_7_7_10_25 471.2 5 4.9 471.2 5 1.3 471.2 5 0.6 471.2 5 0.8 471.2 5 2.5
T_7_7_10_50 734.6 5 1.7 734.6 5 1.2 734.6 5 0.3 734.6 5 0.4 734.6 5 2.2
T_7_7_10_75 912.4 5 1.7 912.4 5 1.2 912.4 5 0.4 912.4 5 0.3 912.4 5 2.3
T_9_9_10_25 730.0 0 2.0 730.0 5 10.5 730.0 5 5.4 730.0 5 12.6 730.0 5 50.0
T_9_9_10_50 1130.4 5 50.5 1130.4 5 1.9 1130.4 5 1.2 1130.4 5 2.9 1130.4 5 132.2
T_9_9_10_75 1478.6 5 16.2 1478.6 5 1.7 1478.6 5 1.1 1478.6 5 1.2 1478.6 5 49.0
Average 721.8 9.9 2.0 721.8 2.4 721.8 1.0 721.8 2.1 721.8 27.1
Total 40 45 45 45 45
Table 4: Results comparing the formulations for the small toroidal instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
R_25_33_10_25 317.2 5 1.4 317.2 5 1.1 317.2 5 0.0 317.2 5 0.1 317.2 5 1.8
R_25_33_10_50 525.8 5 1.4 525.8 5 1.1 525.8 5 0.0 525.8 5 0.1 525.8 5 1.8
R_25_33_10_75 702.8 5 1.4 702.8 5 1.1 702.8 5 0.0 702.8 5 0.1 702.8 5 1.4
R_25_69_10_25 243.0 5 1.5 243.0 5 1.2 243.0 5 0.2 243.0 5 0.1 243.0 5 1.9
R_25_69_10_50 389.8 5 1.5 389.8 5 1.2 389.8 5 0.2 389.8 5 0.2 389.8 5 1.9
R_25_69_10_75 597.2 5 1.5 597.2 5 1.2 597.2 5 0.2 597.2 5 0.1 597.2 5 1.9
R_25_204_10_25 106.8 5 1.8 106.8 5 1.6 106.8 5 1.7 106.8 5 1.0 106.8 5 2.2
R_25_204_10_50 193.0 5 1.7 193.0 5 1.5 193.0 5 1.3 193.0 5 0.8 193.0 5 2.2
R_25_204_10_75 312.2 5 1.7 312.2 5 1.5 312.2 5 1.4 312.2 5 0.7 312.2 5 2.2
R_50_85_10_25 583.2 5 2.7 583.2 5 1.2 583.2 5 0.2 583.2 5 0.3 583.2 5 2.1
R_50_85_10_50 957.2 5 1.9 957.2 5 1.2 957.2 5 0.2 957.2 5 0.4 957.2 5 2.0
R_50_85_10_75 1229.8 5 1.7 1229.8 5 1.2 1229.8 5 0.2 1229.8 5 0.3 1229.8 5 2.0
R_50_232_10_25 355.2 5 92.8 355.2 5 2.1 355.2 5 5.7 355.2 5 3.4 355.2 5 4.9
R_50_232_10_50 657.2 5 38.6 657.2 5 1.8 657.2 5 4.9 657.2 5 3.1 657.2 5 4.3
R_50_232_10_75 889.6 5 35.4 889.6 5 1.8 889.6 5 4.6 889.6 5 2.6 889.6 5 4.3
R_50_784_10_25 146.4 5 189.5 146.4 5 15.0 146.4 5 39.3 146.4 5 42.7 146.4 5 12.7
R_50_784_10_50 278.2 5 147.0 278.2 5 15.9 278.2 5 33.3 278.2 5 40.8 278.2 5 11.9
R_50_784_10_75 415.6 5 110.3 415.6 5 11.3 415.6 5 32.9 415.6 5 32.1 415.6 5 9.0
R_75_157_10_25 792.0 5 861.1 792.0 5 3.6 792.0 5 1.2 792.0 5 3.0 792.0 5 6.1
R_75_157_10_50 1250.6 5 312.7 1250.6 5 2.9 1250.6 5 1.2 1250.6 5 1.3 1250.6 5 4.2
R_75_157_10_75 1870.6 5 84.4 1870.6 5 1.8 1870.6 5 1.1 1870.6 5 1.1 1870.6 5 15.2
R_75_490_10_25 460.2 0 13.9 461.8 5 31.1 461.8 5 163.9 461.8 5 137.2 461.8 5 74.9
R_75_490_10_50 776.4 0 18.5 797.4 5 24.6 797.4 5 219.9 797.4 5 117.8 797.4 5 82.8
R_75_490_10_75 1212.2 0 12.8 1233.4 5 12.6 1233.4 5 115.7 1233.4 5 53.4 1233.4 5 38.1
R_75_1739_10_25 154.4 0 43.5 158.4 5 159.8 158.4 5 827.8 158.4 5 464.6 158.4 5 185.3
R_75_1739_10_50 295.6 0 38.9 300.4 5 120.1 300.4 5 708.6 300.4 5 339.9 300.4 5 112.7
R_75_1739_10_75 454.2 0 31.0 458.2 5 117.1 458.2 5 658.2 458.2 5 321.7 458.2 5 114.7
Average 598.8 90.1 26.4 600.9 19.9 600.9 104.6 600.9 58.1 600.9 26.1
Total 105 135 135 135 135
Table 5: Results comparing the formulations for the small random instances.

Tables 6-10 summarize the results for the small instances when receiving a warm start. Tables 6 and 7 show that all approaches but DCUT presented similar results to those without a warm start for the small grid instances. DCUT achieved large reductions in average times, what indicates that when using such formulation without a warm start, the solver encountered difficulties in finding good quality feasible solutions. Table 8 shows that, for the hypercube instances, offering a warm start did not allow significant improvements using any of the formulations. On the contrary, one less instance was solved to optimality with CYC. Table 9 shows that, similarly to the grid instances, offering a warm start to DCUT allowed significant reductions on the average times to solve small toroidal instances to optimality. Table 10 shows that, for the small random instances, reasonable reductions were achieved when providing a warm start for all approaches when it comes to the average times, with TCYC and DCUT remaining as the best approaches for this subset of instances. Besides, the warm start enhanced the advantage of TCYC over the other approaches for these instances.

CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
G_5_5_10_25 352.4 5 1.4 352.4 5 1.1 352.4 5 0.1 352.4 5 0.1 352.4 5 1.6
G_5_5_10_50 636.8 5 1.4 636.8 5 1.1 636.8 5 0.1 636.8 5 0.1 636.8 5 1.6
G_5_5_10_75 934.2 5 1.4 934.2 5 1.1 934.2 5 0.0 934.2 5 0.1 934.2 5 1.8
G_7_7_10_25 712.4 5 1.5 712.4 5 1.2 712.4 5 0.3 712.4 5 0.6 712.4 5 2.1
G_7_7_10_50 1324.2 5 1.4 1324.2 5 1.1 1324.2 5 0.2 1324.2 5 0.3 1324.2 5 2.0
G_7_7_10_75 1998.6 5 1.5 1998.6 5 1.2 1998.6 5 0.2 1998.6 5 0.4 1998.6 5 2.1
G_9_9_10_25 1180.8 5 5.9 1180.8 5 1.5 1180.8 5 1.5 1180.8 5 11.0 1180.8 5 3.5
G_9_9_10_50 2192.2 5 2.3 2192.2 5 1.3 2192.2 5 1.4 2192.2 5 1.9 2192.2 5 6.4
G_9_9_10_75 3452.2 5 2.0 3452.2 5 1.2 3452.2 5 0.9 3452.2 5 1.4 3452.2 5 2.8
Average 1420.4 2.1 1420.4 1.2 1420.4 0.5 1420.4 1.8 1420.4 2.7
Total 45 45 45 45 45
Table 6: Results comparing the formulations with warm start for the small square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
GNQ_8_3_10_25 319.2 5 1.4 319.2 5 1.1 319.2 5 0.0 319.2 5 0.1 319.2 5 1.8
GNQ_8_3_10_50 579.2 5 1.4 579.2 5 1.1 579.2 5 0.1 579.2 5 0.1 579.2 5 1.8
GNQ_8_3_10_75 838.0 5 1.3 838.0 5 1.1 838.0 5 0.0 838.0 5 0.1 838.0 5 1.6
GNQ_9_6_10_25 786.8 5 1.7 786.8 5 1.2 786.8 5 0.2 786.8 5 2.0 786.8 5 1.9
GNQ_9_6_10_50 1438.6 5 1.5 1438.6 5 1.2 1438.6 5 0.3 1438.6 5 0.7 1438.6 5 2.1
GNQ_9_6_10_75 2195.6 5 1.5 2195.6 5 1.2 2195.6 5 0.3 2195.6 5 0.6 2195.6 5 2.2
GNQ_12_6_10_25 1071.2 5 2.5 1071.2 5 1.5 1071.2 5 1.2 1071.2 5 6.7 1071.2 5 11.9
GNQ_12_6_10_50 1944.2 5 1.8 1944.2 5 1.2 1944.2 5 1.0 1944.2 5 2.2 1944.2 5 4.0
GNQ_12_6_10_75 3022.4 5 1.6 3022.4 5 1.2 3022.4 5 0.7 3022.4 5 1.9 3022.4 5 3.5
Average 1355.0 1.6 1355.0 1.2 1355.0 0.4 1355.0 1.6 1355.0 3.4
Total 45 45 45 45 45
Table 7: Results comparing the formulations with warm start for the small non-square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
I_4_10_25 147.0 5 1.4 147.0 5 1.1 147.0 5 0.0 147.0 5 0.1 147.0 5 1.8
I_4_10_50 289.4 5 1.4 289.4 5 1.1 289.4 5 0.0 289.4 5 0.0 289.4 5 1.8
I_4_10_75 344.0 5 1.4 344.0 5 1.1 344.0 5 0.0 344.0 5 0.0 344.0 5 1.8
I_5_10_25 278.4 5 2.2 278.4 5 1.3 278.4 5 0.8 278.4 5 0.7 278.4 5 2.7
I_5_10_50 545.0 5 1.5 545.0 5 1.2 545.0 5 0.2 545.0 5 0.2 545.0 5 1.9
I_5_10_75 787.4 5 1.5 787.4 5 1.2 787.4 5 0.1 787.4 5 0.1 787.4 5 1.9
I_6_10_25 527.2 2 2296.9 2.0 527.2 5 22.6 527.2 5 60.7 527.2 5 66.3 527.2 5 1764.0
I_6_10_50 975.6 5 19.6 975.6 5 2.7 975.6 5 4.8 975.6 5 4.1 975.6 5 45.8
I_6_10_75 1250.8 5 7.1 1250.8 5 1.4 1250.8 5 1.0 1250.8 5 0.7 1250.8 5 3.2
Average 571.6 259.2 2.0 571.6 3.7 571.6 7.5 571.6 8.0 571.6 202.7
Total 42 45 45 45 45
Table 8: Results comparing the formulations with warm start for the small hypercube instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
T_5_5_10_25 232.8 5 1.4 232.8 5 1.1 232.8 5 0.1 232.8 5 0.1 232.8 5 1.8
T_5_5_10_50 357.2 5 1.4 357.2 5 1.2 357.2 5 0.0 357.2 5 0.1 357.2 5 1.8
T_5_5_10_75 448.6 5 1.4 448.6 5 1.1 448.6 5 0.0 448.6 5 0.1 448.6 5 1.8
T_7_7_10_25 471.2 5 4.6 471.2 5 1.3 471.2 5 0.5 471.2 5 0.7 471.2 5 2.2
T_7_7_10_50 734.6 5 1.7 734.6 5 1.3 734.6 5 0.3 734.6 5 0.4 734.6 5 2.1
T_7_7_10_75 912.4 5 1.6 912.4 5 1.2 912.4 5 0.3 912.4 5 0.4 912.4 5 2.2
T_9_9_10_25 730.0 0 1.7 730.0 5 10.2 730.0 5 5.0 730.0 5 17.9 730.0 5 4.0
T_9_9_10_50 1130.4 5 41.4 1130.4 5 1.6 1130.4 5 1.0 1130.4 5 1.4 1130.4 5 3.6
T_9_9_10_75 1478.6 5 12.7 1478.6 5 1.5 1478.6 5 1.1 1478.6 5 1.2 1478.6 5 3.6
Average 721.8 8.3 1.7 721.8 2.3 721.8 0.9 721.8 2.5 721.8 2.6
Total 40 45 45 45 45
Table 9: Results comparing the formulations with warm start for the small toroidal instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
R_25_33_10_25 317.2 5 1.4 317.2 5 1.1 317.2 5 0.0 317.2 5 0.1 317.2 5 1.8
R_25_33_10_50 525.8 5 1.4 525.8 5 1.1 525.8 5 0.0 525.8 5 0.1 525.8 5 1.8
R_25_33_10_75 702.8 5 1.4 702.8 5 1.1 702.8 5 0.0 702.8 5 0.1 702.8 5 1.4
R_25_69_10_25 243.0 5 1.4 243.0 5 1.2 243.0 5 0.2 243.0 5 0.1 243.0 5 1.8
R_25_69_10_50 389.8 5 1.5 389.8 5 1.2 389.8 5 0.2 389.8 5 0.1 389.8 5 1.9
R_25_69_10_75 597.2 5 1.5 597.2 5 1.2 597.2 5 0.1 597.2 5 0.1 597.2 5 1.9
R_25_204_10_25 106.8 5 1.7 106.8 5 1.5 106.8 5 1.7 106.8 5 0.9 106.8 5 2.3
R_25_204_10_50 193.0 5 1.6 193.0 5 1.4 193.0 5 1.3 193.0 5 0.7 193.0 5 2.3
R_25_204_10_75 312.2 5 1.6 312.2 5 1.4 312.2 5 1.3 312.2 5 0.7 312.2 5 2.2
R_50_85_10_25 583.2 5 2.1 583.2 5 1.2 583.2 5 0.1 583.2 5 0.4 583.2 5 1.9
R_50_85_10_50 957.2 5 1.8 957.2 5 1.2 957.2 5 0.2 957.2 5 0.3 957.2 5 2.1
R_50_85_10_75 1229.8 5 1.6 1229.8 5 1.2 1229.8 5 0.2 1229.8 5 0.3 1229.8 5 2.0
R_50_232_10_25 355.2 5 77.5 355.2 5 2.0 355.2 5 5.3 355.2 5 3.0 355.2 5 4.0
R_50_232_10_50 657.2 5 28.7 657.2 5 1.7 657.2 5 4.7 657.2 5 2.2 657.2 5 4.2
R_50_232_10_75 889.6 5 27.5 889.6 5 1.7 889.6 5 4.1 889.6 5 2.2 889.6 5 4.1
R_50_784_10_25 146.4 5 231.0 146.4 5 8.4 146.4 5 53.8 146.4 5 32.3 146.4 5 9.8
R_50_784_10_50 278.2 5 172.1 278.2 5 8.5 278.2 5 32.6 278.2 5 15.9 278.2 5 9.0
R_50_784_10_75 415.6 5 133.3 415.6 5 7.8 415.6 5 31.2 415.6 5 13.8 415.6 5 8.2
R_75_157_10_25 792.0 5 804.8 792.0 5 3.1 792.0 5 1.0 792.0 5 2.8 792.0 5 3.0
R_75_157_10_50 1250.6 5 195.9 1250.6 5 2.2 1250.6 5 0.9 1250.6 5 1.1 1250.6 5 3.4
R_75_157_10_75 1870.6 5 62.8 1870.6 5 1.7 1870.6 5 1.2 1870.6 5 0.9 1870.6 5 5.8
R_75_490_10_25 461.8 0 13.7 461.8 5 12.5 461.8 5 129.1 461.8 5 69.2 461.8 5 41.4
R_75_490_10_50 797.4 0 15.6 797.4 5 20.1 797.4 5 167.6 797.4 5 70.7 797.4 5 56.7
R_75_490_10_75 1233.4 0 9.8 1233.4 5 6.3 1233.4 5 81.5 1233.4 5 32.2 1233.4 5 35.4
R_75_1739_10_25 158.4 0 47.2 158.4 5 133.0 158.4 5 808.2 158.4 5 372.0 158.4 5 138.7
R_75_1739_10_50 300.4 0 38.3 300.4 5 100.0 300.4 5 690.5 300.4 5 331.5 300.4 5 103.4
R_75_1739_10_75 458.2 0 36.2 458.2 5 97.7 458.2 5 695.4 458.2 5 298.6 458.2 5 91.5
Average 600.9 83.5 26.8 600.9 15.6 600.9 100.5 600.9 46.4 600.9 20.1
Total 105 135 135 135 135
Table 10: Results comparing the formulations with warm start for the small random instances.

The plots in Figure 2 compare the average computation times over all individual small instances in each class for each formulation, (a) using the standalone formulations and (b) running with feasible warm starts generated by the matheuristic MILS- described in Section 3.6 of MelQueRib21. The plots show that TCYC is, overall, the best performing formulation for this set of instances as it takes lower average computation times to solve the instances to optimality. Note that TCYC outperforms all other formulations for hypercube and random graphs. They also illustrate that formulation DCUT is the mostly benefited by the warm starts.

(a) Standalone formulations (b) Running with warm starts
Figure 2: Average computation times in seconds over all small instances in each class for each formulation, (a) first using the standalone formulations, and (b) then running with feasible warm starts generated by the matheuristic MILS-. For each instance, the computation time considered in the averages is either the time to prove optimality (if less than the imposed time limit of 3600 seconds) or this imposed time limit.

5.4 Large instances

Tables 11-20 summarize the results for the large instances, with Tables 11-15 corresponding to the formulations as standalone approaches and Tables 16-20 to the formulations when offered warm starts. Tables 11 and 12 show that for the large grid instances, FLOW clearly outperformed the other approaches, achieving 50 and 54 solved instances for the square and nonsquare grid instances, respectively. Table 13 displays that, for the large hypercube instances, MTZ outperforms the other approaches in terms of number of solutions solved to optimality. This is partly due to the fact that the solver is able to generate good quality feasible solutions earlier in the search process. Table 14 shows that, for the large toroidal instances, FLOW outperformed the other approaches when it comes to the number of solved instances, closely followed by MTZ. Table 15 exhibits that, for the large random instances, MTZ, TCYC, and DCUT solved nearly the same number of instances with TCYC presenting much lower average times. Note that TCYC performed considerably better for the instances with 100 nodes. For very large random instance groups, the open gap was usually very high, with DCUT often achieving the lowest values.

Tables 16-20 show the results for the large instances when a good quality initial feasible solution obtained by the matheuristic of MelQueRib21 is offered to the solver. Tables 16 and 17 reveal that, for the large grid instances, considerable reductions in the average open gaps were achieved for CYC, TCYC and DCUT when compared to the standalone formulation. Table 18 unveils that, for the large hypercube instances, considerable gains in the gaps can be observed for CYC, TCYC and DCUT. Furthermore, the use of good quality warm starts turns DCUT into a very competitive approach for proving optimality of the instances. Table 19 shows that, for the large toroidal instances, again, warm starts allowed a considerable reduction in the average open gaps for CYC, TCYC, and DCUT. Table 20 displays that, for the large random instances, TCYC presented outstanding results when compared to the other approaches, especially on the average time to solve instances to optimality. Even though MTZ proved optimality of one additional instance, the average times achieved by TCYC and DCUT were much lower. We note, though, that for the larger instances, the open gap is still large, even for the best performing approaches.

CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
G_10_10_10_25 1504.0 5 128.5 1504.0 5 3.4 1504.0 5 4.5 1504.0 5 22.7 1504.0 5 795.4
G_10_10_10_50 2722.0 5 5.0 2722.0 5 1.5 2722.0 5 5.0 2722.0 5 10.7 2722.0 5 787.3
G_10_10_10_75 4307.6 5 7.6 4307.6 5 2.5 4307.6 5 5.7 4307.6 5 14.3 4307.6 5 1602.4
G_14_14_10_25 2989.8 0 3.8 2996.2 5 1370.3 2996.2 5 64.8 2996.2 4 1632.1 0.1 1246.0 0 58.6
G_14_14_10_50 5487.4 0 1.8 5489.0 4 55.5 0.1 5489.0 5 166.7 5489.0 5 359.7 953.8 0 82.7
G_14_14_10_75 8794.6 0 3.1 8807.8 1 149.8 0.3 8808.2 5 129.8 8808.2 3 472.5 0.2 3950.8 0 55.4
G_17_17_10_25 4387.0 0 5.8 4427.2 0 0.2 4427.6 4 387.3 0.0 4427.6 1 2691.5 0.1 89.8 0 98.0
G_17_17_10_50 8144.2 0 3.5 8185.6 5 911.8 8185.6 4 669.0 0.1 8185.6 4 1281.0 0.1 192.6 0 97.6
G_17_17_10_75 13125.0 0 4.0 13151.2 1 2497.6 0.4 13155.2 5 786.6 13155.2 1 676.6 0.1 86.6 0 99.3
G_20_20_10_25 5973.4 0 8.9 6154.4 0 0.6 6172.0 2 2280.2 0.1 6173.8 0 0.1 404.8 0 93.5
G_20_20_10_50 11117.6 0 5.8 11343.0 0 0.3 11345.0 2 2933.0 0.1 11345.6 0 0.1 37.2 0 99.7
G_20_20_10_75 18058.6 0 6.3 18388.8 0 0.6 18408.8 2 1623.5 0.0 18408.6 0 0.1 79.2 0 99.6
G_23_23_10_25 7790.8 0 10.5 8099.2 0 1.2 8175.8 1 668.3 0.1 8176.0 0 0.1 858.0 0 89.5
G_23_23_10_50 14762.8 0 6.9 15194.2 0 0.4 15197.6 0 0.1 15195.6 0 0.2 151.4 0 99.0
G_23_23_10_75 23476.0 0 8.9 24348.0 0 1.1 24508.8 0 0.1 24498.4 0 0.2 504.0 0 97.9
Average 8842.7 47.1 5.8 9007.9 624.1 0.5 9026.9 748.0 0.1 9026.2 795.7 0.1 1139.2 1061.7 89.2
Total 15 31 50 33 15
Table 11: Results comparing the formulations for the large square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
GNQ_13_7_10_25 1347.8 5 20.2 1347.8 5 2.0 1347.8 5 5.6 1347.8 5 27.8 1345.4 3 592.2 0.9
GNQ_13_7_10_50 2456.6 5 2.4 2456.6 5 1.2 2456.6 5 1.3 2456.6 5 2.1 2456.6 5 362.3
GNQ_13_7_10_75 3876.6 5 4.1 3876.6 5 1.6 3876.6 5 3.8 3876.6 5 16.4 3876.6 5 410.4
GNQ_18_11_10_25 3002.8 0 3.2 3005.0 4 294.5 0.2 3005.0 5 102.0 3005.0 4 1141.9 0.2 1330.6 0 55.6
GNQ_18_11_10_50 5504.6 0 2.4 5510.2 5 189.1 5510.2 5 160.4 5510.2 5 354.7 1567.6 0 71.4
GNQ_18_11_10_75 8824.0 0 2.7 8829.2 3 1065.3 0.4 8829.6 5 206.3 8829.6 4 809.2 0.2 3761.0 0 57.4
GNQ_23_13_10_25 4533.8 0 6.8 4608.8 0 0.2 4608.8 5 484.8 4609.2 0 0.1 1279.8 0 72.3
GNQ_23_13_10_50 8428.6 0 3.5 8453.2 2 2382.5 0.3 8455.2 4 1038.1 0.1 8455.2 4 1693.3 0.1 30.6 0 99.6
GNQ_23_13_10_75 13532.4 0 4.2 13597.6 0 0.4 13599.4 4 961.1 0.1 13599.4 1 1482.1 0.1 65.4 0 99.5
GNQ_26_15_10_25 5866.8 0 7.9 5998.4 0 0.4 6005.8 3 1077.6 0.0 6005.8 0 0.1 563.8 0 90.6
GNQ_26_15_10_50 10971.2 0 4.6 11098.4 1 951.6 0.3 11099.0 3 2736.7 0.1 11099.0 1 1842.5 0.1 620.6 0 94.4
GNQ_26_15_10_75 17554.4 0 6.3 17887.0 0 0.5 17906.4 4 1349.0 0.0 17906.4 1 2292.7 0.1 129.8 0 99.3
GNQ_29_17_10_25 7317.8 0 10.0 7608.8 0 0.5 7626.0 1 938.7 0.1 7626.4 0 0.1 906.6 0 88.1
GNQ_29_17_10_50 13695.0 0 6.7 14052.4 0 0.5 14066.8 0 0.1 14062.0 0 0.2 275.4 0 98.1
GNQ_29_17_10_75 22145.0 0 7.2 22677.0 0 0.9 22763.8 0 0.1 22763.4 0 0.2 842.8 0 96.3
Average 8603.8 8.9 5.5 8733.8 611.0 0.4 8743.8 697.3 0.1 8743.5 966.3 0.2 1270.2 455.0 78.7
Total 15 30 54 35 13
Table 12: Results comparing the formulations for the large non-square grid instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
I_7_10_25 992.2 0 17.1 1037.4 1 3530.6 1.9 1037.4 0 1.9 1037.4 1 2510.8 1.5 1019.6 0 5.0
I_7_10_50 1908.6 0 7.7 1935.6 5 293.3 1935.6 5 315.7 1935.6 5 147.1 1919.0 3 143.9 3.8
I_7_10_75 2532.8 0 3.5 2533.6 5 34.7 2533.6 5 18.6 2533.6 5 9.3 2533.6 5 117.1
I_8_10_25 1954.6 0 21.7 2032.2 0 4.3 2046.8 0 3.9 2064.4 0 2.3 1018.0 0 52.3
I_8_10_50 3456.8 0 18.8 3680.2 0 4.2 3704.6 0 2.7 3710.2 0 1.7 3677.2 0 3.9
I_8_10_75 4258.2 0 22.3 4856.2 0 3.7 4945.6 3 2525.9 1.0 4948.4 4 728.5 0.5 4919.8 0 1.5
I_9_10_25 0.0 0 100.0 3908.6 0 6.5 3910.4 0 6.5 3925.8 0 5.9 1397.8 0 66.9
I_9_10_50 3926.4 0 56.1 2949.4 0 62.5 7252.0 0 5.1 7309.2 0 4.1 4300.4 0 43.0
I_9_10_75 1172.4 0 89.5 0.0 0 100.0 9244.2 0 3.7 9305.0 0 2.7 5302.2 0 45.1
Average 2244.7 37.4 2548.1 1286.2 26.2 4067.8 953.4 3.5 4085.5 848.9 2.7 2898.6 130.5 27.7
Total 0 11 13 15 8
Table 13: Results comparing the formulations for the large hypercube instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
T_10_10_10_25 943.2 0 3.3 946.2 5 25.3 946.2 5 13.5 946.2 5 32.4 946.2 5 415.5
T_10_10_10_50 1350.4 5 1289.7 1350.4 5 6.1 1350.4 5 4.2 1350.4 5 15.4 1349.4 4 707.5 0.9
T_10_10_10_75 1778.0 5 76.3 1778.0 5 3.0 1778.0 5 1.5 1778.0 5 1.9 1766.0 4 124.6 4.2
T_14_14_10_25 1803.4 0 6.8 1839.4 1 50.7 0.5 1840.6 5 422.0 1840.6 5 647.7 1463.8 1 29.0 25.3
T_14_14_10_50 2794.2 0 2.4 2801.6 3 128.8 0.4 2801.6 5 57.4 2801.6 5 87.0 641.6 0 76.7
T_14_14_10_75 3726.8 1 824.4 1.2 3728.2 5 84.7 3728.2 5 54.7 3728.2 5 74.4 239.0 0 93.7
T_17_17_10_25 2625.8 0 9.2 2732.4 0 0.8 2739.8 2 867.5 0.2 2739.8 1 518.4 0.2 100.6 0 96.4
T_17_17_10_50 4052.4 0 4.1 4112.2 0 0.3 4112.4 5 603.2 4112.4 5 644.7 26.4 0 99.4
T_17_17_10_75 5458.4 0 1.9 5477.6 2 1885.4 0.3 5477.8 5 613.2 5477.8 5 349.7 25.6 0 99.5
T_20_20_10_25 3566.4 0 11.5 3729.8 0 2.5 3801.4 1 3598.0 0.2 3801.0 0 0.2 202.2 0 94.7
T_20_20_10_50 5451.4 0 5.9 5605.2 0 0.8 5622.2 3 1874.8 0.1 5622.0 3 2374.0 0.1 132.2 0 97.7
T_20_20_10_75 7254.0 0 3.6 7376.6 0 0.4 7383.8 5 1297.7 7383.8 4 817.1 0.1 49.8 0 99.3
T_23_23_10_25 4578.8 0 13.9 4852.4 0 3.7 5002.6 0 0.4 5000.0 0 0.4 35.8 0 99.3
T_23_23_10_50 7160.0 0 8.0 7513.0 0 1.0 7552.2 1 2292.9 0.1 7551.6 1 2584.3 0.1 31.0 0 99.6
T_23_23_10_75 9858.6 0 4.3 10090.0 0 0.7 10116.2 2 3357.0 0.0 10116.2 1 1409.0 0.1 67.8 0 99.3
Average 4160.1 730.1 5.9 4262.2 312.0 1.0 4283.6 1075.5 0.2 4283.3 735.1 0.2 471.8 319.1 77.6
Total 11 26 54 50 14
Table 14: Results comparing the formulations for the large toroidal instances.
CYC TCYC FLOW MTZ DCUT
Instance avg opt time gap avg opt time gap avg opt time gap avg opt time gap avg opt time gap
group (s) (%) (s) (%) (s) (%) (s) (%) (s) (%)
R_100_247_10_25 990.8 0 8.4 1004.6 5 16.5 1004.6 5 8.8 1004.6 5 7.7 1004.6 5 27.6
R_100_247_10_50 1683.0 0 7.0 1696.4 5 88.4 1696.4 5 9.1 1696.4 5 12.0 1696.4 5 121.7
R_100_247_10_75 2440.0 0 5.9 2449.2 5 80.0 2449.2 5 8.9 2449.2 5 7.3 2449.2 5 56.5
R_100_841_10_25 456.2 0 33.4 492.0 5 174.4 492.0 5 2387.5 492.0 5 874.6 492.0 5 1480.3
R_100_841_10_50 839.6 0 37.5 949.8 5 594.3 948.8 3 2166.4 1.9 949.8 5 1096.3 949.8 5 1796.8
R_100_841_10_75 1187.6 0 36.7 1350.8 5 313.3 1350.8 5 1852.2 1350.8 5 1020.7 1350.8 5 1488.8
R_100_3069_10_25 142.4 0 71.5 169.6 5 1050.2 166.6 0 36.0 169.6 5 2826.1 169.6 5 1079.6
R_100_3069_10_50 258.8 0 71.4 326.4 5 728.1 315.2 1 3453.0 32.4 326.4 5 2220.8 326.4 5 702.6
R_100_3069_10_75 397.4 0 71.0 489.8 5 723.4 480.4 1 3305.2 29.8 489.8 5 1905.9 489.8 5 622.4
R_200_796_10_25 788.4 0 65.3 1619.8 0 7.7 1676.6 0 4.9 1681.6 0 3.3 1667.4 0 7.6
R_200_796_10_50 961.2 0 74.3 2720.8 0 7.3 2818.0 0 4.3 2824.0 0 2.5 2735.4 0 10.2
R_200_796_10_75 1721.2 0 64.1 3718.4 0 4.5 3757.6 0 3.9 3764.8 1 2903.8 2.2 3706.0 0 7.5
R_200_3184_10_25 264.0 0 82.9 569.0 0 39.0 593.8 0 41.3 613.6 0 34.5 622.4 0 36.1
R_200_3184_10_50 590.6 0 78.9 1072.2 0 35.9 1122.2 0 39.9 1113.2 0 34.6 1113.4 0 40.5
R_200_3184_10_75 668.2 0 83.3 1597.8 0 32.4 1656.0 0 37.1 1605.0 0 33.2 1666.0 0 33.5
R_200_12139_10_25 113.8 0 91.1 151.0 0 82.5 183.0 0 86.1 185.8 0 80.5 189.2 0 62.1
R_200_12139_10_50 212.8 0 90.2 262.4 0 79.9 335.2 0 82.8 339.0 0 78.4 355.6 0 56.1
R_200_12139_10_75 304.0 0 89.7 292.4 0 85.5 471.2 0 83.7 483.6 0 77.2 489.4 0 56.0
R_300_1644_10_25 525.2 0 81.6 1886.6 0 11.5 1930.6 0 10.6 1948.0 0 9.4 1937.6 0 11.9
R_300_1644_10_50 327.8 0 93.7 3273.6 0 17.3 3372.0 0 16.1 3450.6 0 13.4 3386.0 0 15.8
R_300_1644_10_75 482.2 0 94.2 4904.6 0 16.0 5110.4 0 13.9 5146.6 0 12.3 5058.4 0 17.7
R_300_7026_10_25 280.8 0 88.4 566.0 0 59.3 659.8 0 60.2 637.2 0 56.4 650.0 0 55.9
R_300_7026_10_50 459.6 0 90.1 1081.4 0 58.4 1267.2 0 59.8 1270.6 0 54.5 1232.8 0 56.3
R_300_7026_10_75 658.2 0 90.0 1675.4 0 55.2 1757.4 0 60.6 1752.8 0 55.0 1770.0 0 55.1
R_300_27209_10_25 116.0 0 94.9 0.0 0 100.0 0.0 0 100.0 184.2 0 90.7 187.0 0 74.9
R_300_27209_10_50 181.4 0 94.8 5.8 0 99.7 143.6 0 95.9 371.2 0 87.8 349.8 0 68.3
R_300_27209_10_75 290.0 0 95.5 108.4 0 97.0 122.8 0 98.2 551.4 0 90.1 439.6 0 78.2
R_400_2793_10_25 187.6 0 94.9 1996.6 0 23.0 2035.8 0 23.0 2142.0 0 18.9 2130.6 0 19.6
R_400_2793_10_50 842.0 0 89.9 3640.0 0 29.8 3734.2 0 29.3 4113.0 0 22.0 3924.8 0 29.0
R_400_2793_10_75 560.4 0 94.5 5130.0 0 27.4 5219.0 0 27.8 5611.6 0 22.1 5271.2 0 27.3
R_400_12369_10_25 0.0 0 100.0 493.2 0 72.7 699.2 0 74.0 700.0 0 68.5 696.0 0 65.1
R_400_12369_10_50 218.6 0 95.5 654.4 0 77.9 1262.0 0 69.5 1243.2 0 59.8 1230.6 0 59.9
R_400_12369_10_75 310.2 0 96.4 888.2 0 80.0 1819.6 0 72.1 1832.6 0 64.8 1805.6 0 62.7
R_400_48279_10_25 83.8 0 97.5 16.4 0 99.1 0.0 0 100.0 0.0 0 100.0 195.8 0 92.7
R_400_48279_10_50 116.6 0 97.6 79.2 0 96.9 0.0 0 100.0 0.0 0 100.0 381.8 0 90.0
R_400_48279_10_75 271.0 0 97.4 28.0 0 99.4 0.0 0 100.0 0.0 0 100.0 599.6 0 92.5
R_500_4241_10_25 0.0 0 100.0 2228.0 0 28.2 2129.6 0 34.7 2181.2 0 31.5 2300.0 0 28.8
R_500_4241_10_50 0.0 0 100.0 3656.2 0 30.7 3526.8 0 36.4 3617.2 0 33.1 3679.4 0 34.0
R_500_4241_10_75 0.0 0 100.0 5426.6 0 33.7 5403.8 0 37.5 5466.4 0 34.7 5658.2 0 33.4
R_500_19211_10_25 0.0 0 100.0 119.8 0 94.8 0.0 0 100.0 731.0 0 78.3 730.6 0 76.0
R_500_19211_10_50 0.0 0 100.0 413.2 0 87.5 258.6 0 94.5 1280.8 0 72.4 1264.6 0 70.0
R_500_19211_10_75 0.0 0 100.0 301.8 0 95.0 0.0 0 100.0 1975.2 0